ANALYSIS ON THE EFFECTS OF WALL PROPERTIES ON MHD PERISTALTIC FLOW OF A DUSTY FLUID THROUGH A POROUS MEDIUM

W2198828488 · 수집 2026-10-04 18:12:22

T12063 Heat and Mass Transfer in Porous Media

CoT 라벨링

라벨링 전

PDF 파일

자동 다운로드한 PDF PDF 보기

기본 정보

DOI 제목 발행년도 저널 URL 페이지 열기 PDF URL PDF 열기 발행기관 피인용수 라이선스

추가 정보 초록

In this paper, a MHD peristaltic transport of dusty fluid through a porous medium in a uniform channel with elastic wall properties has been considered.This phenomenon is modeled on a system of partial differential equations governing the motion of the fluid and solid particles.This system of Navier stokes equations is linearized under long wavelength approximation and solved with appropriate boundary conditions using perturbation technique.The analytical solution is obtained for streamline functions and axial velocity profiles for both fluid and dust particles.The effects of membrane tension parameter, mass characterizing parameter, damping parameter, mass concentration of the dust particles, Hartmann number and permeability of the porous medium on streamline pattern of the fluid is studied.It is observed that the trapped bolus size increases with the increase in tension parameter, Hartmann number and permeability of the porous medium, but decreases with increase in damping and mass concentration parameters.

본문

International Journal of Pure and Applied Mathematics
Volume 102 No. 2 2015, 247-263
ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
# url: http://www.ijpam.eu AP

doi: http://dx.doi.org/10.12732/ijpam.v102i2.7 ijpam.eu

ANALYSIS ON THE EFFECTS OF WALL PROPERTIES ON

MHD PERISTALTIC FLOW OF A DUSTY FLUID

THROUGH A POROUS MEDIUM

Siva Parthasarathy 1 §, Govindarajan Arunachalam 2, M. Vidhya 3

1,2Department of Mathematics
SRM University
Kattankulathur, 603203, INDIA

3Department of Mathematics
Sathyabama University
Sholinganallur, 603203, INDIA

Abstract: In this paper, a MHD peristaltic transport of dusty fluid through
a porous medium in a uniform channel with elastic wall properties has been
considered. This phenomenon is modeled on a system of partial differential
equations governing the motion of the fluid and solid particles. This system of
Navier stokes equations is linearized under long wavelength approximation and
solved with appropriate boundary conditions using perturbation technique. The
analytical solution is obtained for streamline functions and axial velocity profiles
for both fluid and dust particles. The effects of membrane tension parameter,
mass characterizing parameter, damping parameter, mass concentration of the
dust particles, Hartmann number and permeability of the porous medium on
streamline pattern of the fluid is studied. It is observed that the trapped bolus
size increases with the increase in tension parameter, Hartmann number and
permeability of the porous medium, but decreases with increase in damping
and mass concentration parameters.

⃝c 2015 Academic Publications, Ltd.
Received: February 20, 2015

url: www.acadpubl.eu

§ Correspondence author

248 S. Parthasarathy, G. Arunachalam, M. Vidhya

AMS Subject Classification: 76S05, 76T20, 74F10, 35K41
Key Words: wall properties, MHD, peristaltic flow, porous medium, trapping, reflux

1. Introduction

Peristalsis is the movement of body fluids travelling through tubular organs
continuously by contraction and relaxation due to change in pressure along the
walls. This phenomenon is observed in the human body during the transport
of lymph on lymphatic vessels, while swallowing food through the esophagus to
the stomach, the vasomotion of small blood vessels, transport of chyme through
the small intestine, the passage of urine from the kidneys through the urethra
to the urinary bladder, the flow of semen in the male reproductive tract, and
the movement of ovum in the female fallopian tube. Peristaltic phenomenon
is used in biomedical instruments like dialysis machines, heart-lung machines,
artificial heart and ortho machines, and transport of toxic material waste inside
the sanitary ducts. Peristaltic concept is used in nuclear industry to avoid
contamination of the outside environment.

Beginning with the first investigation of Latham [2], several theoretical and
experimental attempts have been made to understand peristaltic action in different situations. The theory of trapping and reflux was first developed by
Shapiro et al. [3]. Shapiro discussed the theoretical results for both plane and
axisymmetric geometries. Srinivasa Rao [19] studied the peristaltic transport
of non uniform channel under the long wavelength approximation. Muthu et
al. [8] studied the influence of wall properties in the peristaltic motion of a
micropolar fluid. Mochtar et al. [9] developed a new model to study the effect of wall properties on peristaltic transport of a viscous fluid. Influence of
wall properties on peristaltic transport with heat transfer was studied by Radhakrishnamacharya et.al [11]. Hina et al. [17] discussed the effects of wall
properties on peristaltic motion with respect to heat/mass transfer. Muthu et
al. [12] considered peristaltic motion of micropolar fluid in circular cylindrical
tubes. Sankad et al. [14] analyzed the effect of wall properties on peristaltic
transport of micropolar fluid under long wavelength approximation. Few others
have discussed the trapping and reflux for different kind of fluids. Hayat et al.

[16] analyzed the effect of wall properties on the peristaltic flow of a third grade
fluid in a curved channel with heat and mass transfer. Dheja et al. [20], Nadeem
et al. [21] studied the trapping and reflux for Jeffery fluid. Sankad et al. [23]
have discussed the peristaltic flow of Hershel- Bulkley fluid. The MHD flow of a

ANALYSIS ON THE EFFECTS OF WALL PROPERTIES ON... 249

fluid in a channel with elastic rhythmically contracting walls (peristaltic flow) is
of interest in connection with certain problems of the movement of conductive
physiological fluids, and with the need for theoretical research on the operation
of a peristaltic MHD compressor. Also the principle of magnetic field may be
used in clinical application (magnetic resonance imaging MRI). Makhimer et
al. [7] discussed the nonlinear peristaltic transport of Magneto-hydrodynamics
(MHD) flow through a porous medium. Effects of magnetic field and wall properties of micropolar fluid on peristaltic flow was discussed by Afifia et al. [15].
Moreover, recently Dheia et al. [20] studied the effects of wall properties and
heat transfer on the peristaltic flow of Jeffry fluid through porous medium. The
two-phase flow plays an important role in recent times in the purification of oil
in the oil industry. Also, such flows occur in fluidization flow in rocket tubes,
blood flow in capillaries, paint spraying confusion etc. Saffmans [1] dusty fluid
model serves as a good model for describing blood as a binary system. Tridib
et al. [4] investigated the influence of wall properties and Poiseuille flow in peristalsis. Kaimal [5], Radhakrishnamacharya [6] studied peristaltic transport of
a solid-fluid mixture through a two dimensional channel. Mohammed Haroun

[10] detailed the effect of wall compliance on peristaltic transport of a Newtonian fluid in an asymmetric channel. Srinivasacharya [13] studied the effects of
wall properties on peristaltic transport of a dusty fluid and explained trapping
and reflux in their article. Nabil Eldabe et al. [18] discussed the wall properties
of peristaltic motion with heat and mass transfer of a dusty fluid through a
horizontal porous channel. The peristaltic flow of viscous fluid in a rectangular
duct with compliant walls has been studied by Nadeem et al. [22].

The primary goal of this article is to construct a mathematical model to
examine the influence of wall properties on unsteady MHD peristaltic flow of
a dusty fluid through the porous medium in a symmetric channel. This phenomenon is modeled on a system of partial differential equations governing the
motion of fluid and solid particles. Problem is formulated in Section 2. Section
3 includes the solutions for the problem by using perturbation method. Numerical results and discussion are given in Section 4. The conclusions have been
summarized in Section 5.

2. Formulation of the Problem

We consider the peristaltic transport of an incompressible fluid that contains
small solid particles, whose number density N (assumed to be constant) is large
enough to define average properties of the dust particles at a point through a

250 S. Parthasarathy, G. Arunachalam, M. Vidhya

Geometry of 2- dimensional peristaltic motion of channel walls

symmetrical 2-dimensional channel through porous medium in the presence of
transverse magnetic field. The half width of the channel is d. The waves on
the channel walls have speed c, and amplitude phase a. The wave length of the
channel is λ. We select Cartesian coordinates (x,y) with x being in the direction
of wave propagation and y normal to it. η is the displacement of upper wall, -η
the displacement of lower wall and t the time. The geometry of the wall surface
is described by

�X¯ - ct¯� � (1)

Y = ±η = ±

d + a sin 2π

The equations describing two dimensional flow are: For fluid particles:

∂u¯

(2)

∂x¯ + ∂ ∂ y v¯ ¯ = 0

- ∂∂u¯t ¯ + ¯u∂ ∂x u ¯ ¯

+ KN0

KN0 ( ¯up − u¯) − ν

ρ K

u∂ ∂x u ¯ ¯ + ¯v∂ ∂ u y ¯ ¯

∂x ¯2 + ∂ ∂ 2 y ¯u¯2

- ∂2u¯

∂y ¯2

∂y¯

= − 1

∂p¯
∂x¯ + ν

u¯ − σB2

2 u¯ (3)

- ∂∂v¯t ¯ + ¯u∂ ∂x v ¯ ¯

∂x ¯2 + ∂ ∂ 2 y ¯v¯2

- ∂2v¯

u∂ ∂x v ¯ ¯ + ¯v∂ ∂ v y ¯ ¯

∂y¯

= − 1

∂p¯
∂y¯ + ν

ANALYSIS ON THE EFFECTS OF WALL PROPERTIES ON... 251

KN0 ( ¯vp − v¯) − ν

ρ K

+ KN0

v¯ (4)

For dust particles

∂u¯p

(5)

∂ ∂ v y ¯ ¯ p = 0

∂∂ux¯¯p + ∂ ∂ v y ¯ ¯ p

∂t ¯ + u¯p

- ∂u¯p

∂t ¯ + u¯p

- ∂v¯p

∂u¯p

∂x¯ + v¯p

∂u¯p

= K m (¯u − u¯p) (6)

= K

∂v¯p

∂x¯ + v¯p

∂v¯p

∂u¯p

∂y¯

∂v¯p

∂y¯

= K (7)

m (¯v − v¯p)

= K

With the appropriate boundary conditions ¯u = 0, u¯p = 0 at y¯ = ±η¯ Where
u,¯ ¯v is the velocity of the fluid particles and u¯p, v¯p is the velocity of the dust
particles, p¯ - Fluid pressure, B0 - Transverse magnetic field, σ - Coefficient of
electric conductivity, ρ - Density of the fluid, ν - Kinematic coefficient of the
viscosity of fluid, K - Resistance coefficient for the dust particles(a constant),
K1 - Permeability of the porous medium, m - Mass of the solid particles. The
governing equation of motion of the flexible wall may be expressed as

L (¯η) = p¯ − p¯0 (8)

where L is an operator that is used to represent the motion of the stretched
membrane with damping forces such that

∂ ∂ x ¯22 + m ′ ∂ ∂ t ¯22

L = −T ∂ ∂ x ¯2

∂ ∂ t ¯22 + C ∂ ∂

(9)
∂ ∂ t ¯

T-Tension in the membrane, m-Mass per unit area, C - Coefficient of viscous
damping forces and p¯0 -Pressure on the outside of the wall due to tension in
the muscles. With the appropriate boundary conditions,From equation (3)

- ∂∂u¯t ¯ + ¯u∂ ∂ u x ¯ ¯

∂

∂ ∂ x p ¯ ¯ = ρν

∂x¯ (L (¯η)) = ∂ ∂ x p ¯ ¯

∂x ¯2 + ∂ ∂ 2 y ¯u¯2

- ∂2u¯

∂y ¯2

∂y¯

- ρ

∂ u x ¯ ¯ + ¯v ∂ ∂ u y ¯ ¯

+ KN0

KN0 ( ¯up − u¯) − ν

ρ K

u¯ − σB2

2 u¯ (10)

Introducing the stream function ψ ¯, φ ¯ such that

∂ ∂ ψ y ¯ ¯, ¯v = ∂ ∂ ψ x ¯ ¯

∂ ∂ φ y ¯ ¯, v¯p = ∂ ∂ φ x¯ ¯

∂x¯

u¯ = ∂ ∂ ψ y ¯ ¯

∂ ∂ ψ x ¯ ¯, u¯p = ∂ ∂ φ y ¯ ¯

252 S. Parthasarathy, G. Arunachalam, M. Vidhya

and the following non-dimensional variables

φ¯
Kd 2 /m ,

x λ ¯ , y = d y¯

η d ¯, t = ν λ t¯

ν λ t¯, ψ =

x = x λ ¯

d y¯, η = η d ¯

ψ¯
ν , φ =

into equations (2) - (7) we obtain the following equations (after eliminating the
pressure term)

∂t

- ∂

�∇ 2 1

2 1 ψ + ∂ψ

∂y

�∇ 2 1

2 1 ψ - ∂ψ

∂x

�∇ 2 1

2 1 ψ

- �

2 1φ −∇1 2 ψ - k 1 0

∂y

∂
∂x

∂
∂y

4 1 ψ + P R ∇ 2 1

- 1

= ∇ 4 1

∇ 2 1

2 1 ψ − M 2 ∂2ψ (11)

∂y 2

�∇ 2 1

2 1 φ + ∂φ

∂y

�∇ 2 1

2 1 φ - ∂φ

∂x

�∇ 2 1

2 1 φ = R∇ 2 1

- �

∂
∂y

where

R ∂

∂t

2 1 =

- ∂X∂2

∂y

∂
∂x

2 1 ψ −∇2 1 φ (12)

-, ǫ = d a

∇ 2 1

∂X 2 + ∂Y ∂2

∂2

∂Y 2

d a , δ = λ d

(13)
λ d

R = Kdνm 2 and P = KN νp 0 d 2

νp 0 d 2 are non-dimensional parameters and k0 = K d 21

Kd 2 νp d 2

B 0 d √ σ
is the coefficient of permeability of porous medium, M 2 = √ µ is the
Hartmann number The boundary conditions in the non dimensional form is

∂ψ∂y = ∂φ ∂y

∂ψ

at y = ±η = ± [1 + ǫ sin 2π (x − t)] (14)

∂φ ∂y = 0

- 1

∂ 2

∂x∂t +

∂ψ

2 1

∂y

∂ψ

∂ψ∂y − ∂ψ ∂x

∂x

∂ψ

∂y

+ P

∂φ∂y − ∂ψ ∂y

∇ 2 1

∂y − δ

∂t

- ∂

∂ψ

∂ψ∂y + ∂ψ ∂y

∂y

∂
∂x

∂
∂y

∂x∂t 2 + E3

η (15)

∂ 3

∂x 3 + E2

∂ 3

- 1

∂y =

- ∂ψ

+ M 2

at y = ±η = ± [1 + ǫ sin 2π (x − t)] where

E1 = Td4

νρ − MembraneTensionParameter,

E2 = m ′ d2

λ 3 ρ − MasscharactarisingParameter,

E3 = Cd3

λ 3 νρ − TheDampingTensionParameter,

ANALYSIS ON THE EFFECTS OF WALL PROPERTIES ON... 253

3. Solution Procedure

Assuming the parameter δ is very small, the stream functions ψ and φ may be
expanded in power series of δ as

ψ = ψ0 + δψ1 + δ 2 ψ2 + . . . . . ., φ = φ0 + δφ1 + δ 2 φ2 + . . . . . . (16)

Substituting (16) in equations (11) - (15) and collecting the coefficients of various powers of δ on both sides, we obtain the following sets of coupled linear
differential equations for ψ0, φ0 and ψ1, φ1

Zeroth order in δ

∂ 4 ψ0

∂y 4 + P

- 1

∂y 2

- L 2 ∂2ψ0 (17)

∂y 2 = 0

∂ 2 φ0

∂∂y 2 φ 2 0 − ∂ ∂y 2ψ20

∂ ∂y 2ψ20 − ∂ ∂y 2φ20

R ∂ ∂y 2ψ20

(18)
∂y 2φ20 = 0

and the corresponding boundary conditions at y = ±η are

∂ψ0

(19)

∂φ ∂y 0 = 0

∂ψ∂y0 = ∂φ ∂y 0

∂ 3 ψ0

∂y 3 + P

- 1

∂φ∂y0 − ∂ψ ∂y 0

∂y

∂ 3

∂φ0

- L 2 ∂ψ0

∂y

η (20)

∂ 3

∂x 3 + E2

∂x∂t 2 + E3

∂x∂t +

∂ 2

First order in δ

∂ 4 ψ1

∂y 4 + P

- 1

∂y 2

- L 2 ∂2ψ1

∂y 2

∂ 2 φ1

∂∂y 2 φ 2 1 − ∂ ∂y 2ψ21

∂ 3 ψ0

∂y 3

(21)

∂t∂y 2 + ∂ψ ∂y 0

- ∂3ψ0

∂y

∂x∂y∂ 3 ψ0 2 − ∂ψ ∂x 0

∂x

∂y

∂y 2ψ21 − ∂ ∂y 2φ21

∂t∂y 3 φ02 + ∂φ ∂y 0

∂ 3 φ0

(22)
∂y 3

R ∂ ∂y 2ψ21

∂ ∂y 2φ21 = R ∂ ∂t∂y 3 φ02

∂x∂y∂ 3 φ0 2 − ∂φ ∂x 0

and the corresponding boundary conditions at y = ±η are

∂ψ∂y1 = ∂φ ∂y 1

∂ψ1

(23)

∂φ ∂y 1 = 0

254 S. Parthasarathy, G. Arunachalam, M. Vidhya

∂ 3 ψ1

∂y 3 + P

- 1

∂φ∂y1 − ∂ψ ∂y 1

∂y

∂φ1

- L 2 ∂ψ1

∂y

= ∂ ∂t∂y 2ψ0

∂y

∂ ∂t∂y 2ψ0 + ∂ψ ∂y 0

∂ 2 ψ0

∂∂x∂ 2 ψ0 − ∂ψ ∂x 0

∂x

(24)
∂y 2

where

L 2 = 1

+ M 2

The solutions of equations (17) and (18), subject to the boundary conditions (19) and (20), are

ψ0 = L A31 sinh cosh Lη Ly

(25)

A L 21

L A31 sinh cosh Lη Ly − A L 21

(26)

φ0 = R

L 3 cosh Lη − A L 21

- A1 sinh Ly

where

A1 = ǫ

�E3 (2π) 2 sin 2π (x − t) − (E1 + E2) (2π) 3 cos 2π (x − t)

(27)

The solutions of equations (21) and (22), subject to the boundary conditions (23) and (24), are

ψ1 = T30y + T31 sinh Ly + T12y cosh Ly + T13 sinh 2Ly + T14y 2 sinh Ly (28)

φ1 = T42y + T38 sinh Ly + T39y cosh Ly + T40 sinh 2Ly + T41y 2 sinh Ly (29)

S = N0m = PR

where S is mass concentration of dust particles, The axial velocities of the fluid
(u) and the dust particle (up) are given by

+δ

T14Ly 2 cosh Ly + T33y sinh Ly

u = A1

L 2

cosh Lη − 1

- cosh Ly

+ δ (T30 + T32 cosh Ly + 2T13 cosh 2Ly)

- (30)

ANALYSIS ON THE EFFECTS OF WALL PROPERTIES ON... 255

up = A1R

L 2

cosh Lη − 1

- cosh Ly

+ δ (T42 + (T38L + T39) cosh Ly)

+δ

2T40L cosh 2Ly + (T39L + 2T41) y sinh Ly + T41Ly 2 cosh Ly (31)

4. Rusults and Discussion

In this section, the behavior of various parameters involved in the expressions
of axial velocity u = ψ0 + δψ1 and steam function ψ are provided. In particular
the variations of elastic tension in the membrane E1, the mass per unit area
E2, the coefficient of viscous damping E3, mass characterizing parameter S,
Hartmann number M and permeability of the porous medium k0 are observed.
Fig. 1 shows the behavior of parameters which is involved in velocity profile.
Fig. 1.a and 1.b depict that the velocity profile increases as the elastic tension
in the membrane and the mass per unit area increases. Because of the elasticity
of the walls, there is less resistance to the flow and so the velocity increases as
noted by (17). Fig. 1.c shows that u increases by increasing E3. Figs. 1.d and
1.e show that the velocity decreases with an increase in mass characterizing
parameter and Hartmann number. Fig. 1.f depicts that the velocity increases
with an increase in the permeability of the porous medium k0.

Trapping is an important phenomenon in peristaltic motion. In a wave
frame, the streamline under particular conditions split to trap a bolus which
moves as a whole with the speed of the peristaltic wave. The effects of various
parameters like elastic tension in the membrane E1, the mass per unit area E2,
the coefficient of viscous damping E3, mass characterizing parameter S, Hartmann number M and permeability of the porous medium k0 on trapping and
reflux were numerically evaluated and the results are graphically presented in
Fig. 2. Figs. 2.a and 2.b shows that the streamline got closer as the tension
parameter (E1) increased and the phenomenon of trapping was observed. Observing figs. 2.a and 2.b it is significant to note that the trapping phenomenon
became predominant and the area of the trapped bolus increased as the tension parameter increased. We observe from figs. 2.c and 2.d that the trapping
phenomenon occurs for mass per unit area (E2) It can be seen from figs 2.c and
2.d that the trapped bolus increases as the mass per unit area E2 increases.
It can be seen figs. 2.e and 2.f that as damping force E3 increased the area
of the trapped bolus in some regions decreased (left side) and in some regions
remained unchanged (right side). The effect of the mass characterizing parameter (S) on the streamline pattern is shown in Figs. 2.g and 2.h. Change in

256 S. Parthasarathy, G. Arunachalam, M. Vidhya

E1 = 0.5

E1 = 0.75

E1 = 1.0

E2 = 0.5

E2 = 0.75

E2 = 1

120

100

0
−1 −0.5 0 0.5 1

(a) Variation of E1 on u when k0
= 1, M = 0.2, S = 0.2, E2 = 0.5,
E3 = 0.6.

(b) Variation of E2 on u when k0
= 1, M = 0.2, S = 0.2, E1 = 0.75,
E3 = 0.6.

S = 0.2
S = 0.3
S = 0.4

E3 = 0.6

E3 = 0.8

E3 = 1.0

0
−1 −0.5 0 0.5 1

(d) Variation of S on u when k0
= 1, M = 0.2, E1 = 0.75, E2 =
0.5, E3 = 0.6.

M = 0.3
M = 0.6
M = 0.9

k0 = 0.5

k0 = 0.75

0
−1 −0.5 0 0.5 1

k0 = 1.0

(e) Variation of M on u when k0
= 1, S = 0.2, E1 = 0.75, E2 =
0.5, E3 = 0.6.

(f) Variation of k0 on u when S
= 0.2, M = 0.2, E1 = 0.75, E2 =
0.5, E3 = 0.6.

Figure 1: velocity profile of the fluid when ǫ = 0.2, δ = 0.2.

ANALYSIS ON THE EFFECTS OF WALL PROPERTIES ON... 257

1.5

0.5

−0.5

−1

−1.5

−2
0 0.1 0.2 0.3 0.4 0.5

(a) E1=0.1

1.5

0.5

−0.5

−1

−1.5

−2
0 0.1 0.2 0.3 0.4 0.5

1.5

0.5

−0.5

−1

−1.5

−2
0 0.1 0.2 0.3 0.4 0.5

(e) E3=0.1

1.5

0.5

−0.5

−1

−1.5

−2
0 0.1 0.2 0.3 0.4 0.5

(b) E1=0.5

1.5

0.5

−0.5

−1

−1.5

−2
0 0.1 0.2 0.3 0.4 0.5

(d) E2=0.5

1.5

0.5

−0.5

−1

−1.5

−2
0 0.1 0.2 0.3 0.4 0.5

(f) E3=0.75

258 S. Parthasarathy, G. Arunachalam, M. Vidhya

1.5

0.5

−0.5

−1

−1.5

−2
0 0.1 0.2 0.3 0.4 0.5

(a) S = 0.1

1.5

0.5

−0.5

−1

−1.5

−2
0 0.1 0.2 0.3 0.4 0.5

1.5

0.5

−0.5

−1

−1.5

−2
0 0.1 0.2 0.3 0.4 0.5

(e) k0 = 0.3

1.5

0.5

−0.5

−1

−1.5

−2
0 0.1 0.2 0.3 0.4 0.5

(b) S = 0.75

1.5

0.5

−0.5

−1

−1.5

−2
0 0.1 0.2 0.3 0.4 0.5

(d) M = 0.2

1.5

0.5

−0.5

−1

−1.5

−2
0 0.1 0.2 0.3 0.4 0.5

(f) k0 = 0.8

Figure 3: Streamline pattern of the fluid particles for ǫ = 0.2, δ = 0.2,
E1 = 0.5, E2 = 0.75, E3 = 0.75, S = 0.5 and M = 0.2, k0 = 1

ANALYSIS ON THE EFFECTS OF WALL PROPERTIES ON... 259

the character of the streamlines with the variation in the mass concentration
of dust particles (S) was significant. The trapping was observed and the area
of the trapped bolus decreases as increase of mass concentration of the dust
particle S.

Figs. 2.i and 2.j shows the influence of Hartmann number M on trapping. It
shows that the size of the left trapped bolus increases on both sides with increase
in M. The influence of porous permeability parameter k0 on the trapping is
analyzed in figs 2.k and 2.l. It showed that the volume of the trapped bolus is
unchanged with increasing permeability parameter k0.

5. Concluding Remarks

The present study deals with the effect of wall properties on MHD peristaltic
flow of a dusty fluid through a porous medium channel. We obtained the
analytical solution of the problem under long wavelength and low Reynolds
number assumptions. The results are analyzed for different values of parameters
namely elastic tension in the membrane E1, the mass per unit area E2, the
coefficient of viscous damping E3, mass characterizing parameter S, Hartmann
number M, permeability of the porous medium k0. The key features of the
present study are

1. The axial velocity increases as the elastic tension in the membrane and
the mass per unit area increases. The velocity increases with an increase
in the permeability of the porous medium k0. the velocity decreases with
an increase in mass characterizing parameter and Hartmann number.

2. The volume of the trapped bolus increases with increase in E1, E2 and
k0. Moreover, more trapped bolus appears with increase in E1 and E2.

3. The volume of the trapped bolus decreases with increase in S.

4. The volume of the left trapped bolus decreases with increase in E3 and
M, whereas it has opposite behavior on the right hand side of the channel.

6. Appendix

T1 = 1

coshLη ∂A ∂t 1 − A1LsinhLη ∂η ∂t

- (coshLη) 2

260 S. Parthasarathy, G. Arunachalam, M. Vidhya

T2 = 1

coshLη ∂A ∂x 1

- (

∂A ∂x 1 − A1LsinhLη ∂η ∂x

∂x
(coshLη) 2

T3 = 2L 2 TcoshLη 2A1 , T4 = − T L 2A2 1

2A1 T5 = T2

L 2 , L 2

T2

L 2,

T6 = −

∂AL∂x 2 1 , T7 = 2coshLη T5A1 , T8 = coshLηT6A1 ,

T9 = S1 (T1 + T4), T10 = S1 (T3 − T7), T11 = −S1T8,

2 T L 93 − 5 4 T L 113 , T13 = 12 T10 L 4

T12 = 2 T L 9

T10

12L 4,

T14 = T11

T11, T17 = T1

4L3 L

L 1 , T18 = −

∂A 1

L∂t 2 ,

T2 T20 = T6, T21 = T22 = − A1

L , L 2 coshLη , L 2

T19 = T2

A1

L 2,

T23 = T21T19, T24 = T20T21 + T22T19, T25 = T22T20,

T5A1 T6A1
T26 = LcoshLη , T27 = LcoshLη ,

T28 = T17 + T24, T29 = T18 + T25, T32 = T31L + T12, T33 = T12L + 2T14,

T30 = 1

L 2

2T12L 2 + 6T14L − T28 coshLη + 6T13L 3 cosh2Lη
��
4T14L 2 + T27

+ 1

L 2

T31 =

4T14L 2 + T27 ηsinhLη − T29 − T23 (coshLη) 2 + T26 (sinhLη) 2�
��

- (T28 + (T12L + 2T14) ηsinhLη) +

T14Lη 2 + T12

LcoshLη

�+R 2 T8,

−R 2 T1−R 2 T4, T35 = R

−� R 2 T3 + R 2 T7, T37 =

T14Lη 2 + T12 coshLη + 2T13Lcosh2Lη

,
LcoshLη

T12L 2 + 4T14L

RT� 14L 2,

T34 = R

T31L 2 + 2T14L + 2T14

- T36 = 4RT13L 2

T36 = 4RT13L 2 - R 2 T3 + R 2 T7, T37 = RT14L 2,

T38 = T L 342

L T335 + T L 374

T L 342 − 2 L T335

T L 374, T39 = T L 352

T L 352 − 2 L T373

L T373 ,

T36 T41 = T37

4L 2, L 2

T40 = T36

37

L 2,

T42 = −

+ (

��T38L + T39 + T41Lη 2�

coshLη

+ (T39Lη + 2T41η) sinhLη + 2T40Lcosh2Lη) .

ANALYSIS ON THE EFFECTS OF WALL PROPERTIES ON... 261

References

[1] P.G.Saffman, the Stability of Laminar flow of a Dusty Gas J. Fluid Mech.,

13 (1962), 122-128.

[2] T.W.Latham, Fluid motion in Peristaltic pump, MS thesis, Mit Cam
bridge, (1966).

[3] A.H.Shapiro, M.Y.Jaffrin and S.L.Weinberg, Peristaltic pumping with long

wavelength at low Reynolds number, Journal of fluid mechanics, 37 (1969),
799-825.

[4] K.Tridib, Mittra and ShyamN.Prasad, On the influence of wall proper
ties and Poiseuille flow in peristalsis, Journal of Biomechanics, 6 (1973),
681693.

[5] M. R. Kaimal, Peristaltic Pumping of a Newtonian Fluid With Particles

Suspended in It at Low Reynolds Number Under Long Wavelength Approximations, Journal of Applied Mechanics, 45 (1978), 32-36.

[6] G. Radhakrishnama Charya, Pulsatile flow of a dusty fluid through a con
stricted channel, Zeitschrift fr angewandte Mathematik und Physik ZAMP,
29 (1978), 217-225.

[7] Kh.S.Mekheimer and T.H.Al-Arabi, Nonlinear peristaltic transport of

MHD flow through a porous medium, IJMMS, 26 (2003), 16631682.

[8] P.Muthu, B.V.Rathish Kumar and Peeyush Chandra and J.Anziam, On

the influence of wall properties in the peristaltic motion of micropolar fluid,
ANZIAM J, 45 (2003), 245260.

[9] A.Mokhtar, Abd Elnaby, Mohamed H. Haroun, A new model for study the

effect of wall properties on peristaltic transport of a viscous fluid, Communications in Nonlinear Science and Numerical Simulation, 13 (2008),
752762.

[10] Mohamed H.Haroun, Effect of wall compliance on peristaltic transport of

a Newtonian fluid in an asymmetric channel, Mathematical Problems in
Engineering, Article ID 61475 (2006), 119.

[11] G. Radhakrishnamacharya, Ch. Srinivasulu, Influence of wall properties

on peristaltic transport with heat transfer, C. R. Mecanique, 335 (2007),
369373.

262 S. Parthasarathy, G. Arunachalam, M. Vidhya

[12] P. Muthu, B.V. Rathish Kumar, Peeyush Chandra, Peristaltic motion of

micropolar fluid in circular cylindrical tubes Effect of wall properties, Applied Mathematical Modelling, 32 (2008), 20192033.

[13] D. Srinivasacharya, G.Radhakrishnamacharya, CH.Srinivasulu, The Ef
fects of Wall Properties on Peristaltic Transport of a Dusty Fluid, Turkish
J. Eng. Env. Sci, 32 (2008), 357 365.

[14] G. C. Sankad, G. Radhakrishnamacharya, Effect of wall properties on peri
staltic transport of micropolar fluid in a non-uniform channel, Int. J. of
Appl. Math and Mech, 6 (2010), 94-107.

[15] N. A. S. Afifia, S. R. Mahmoudb and H. M. Al-Isedec, Effect of Mag
netic Field and Wall Properties on Peristaltic Motion of Micropolar Fluid,
International Mathematical Forum, 6 (2011), 1345 1356.

[16] T. Hayat, S. Hina, Awatif A. Hendi, S. Asghar, Effect of wall properties

on the peristaltic flow of a third grade fluid in a curved channel with heat
and mass transfer, International Journal of Heat and Mass Transfer, 54
(2011), 51265136.

[17] S. Hina, T. Hayat, S. Asghar, Awatif and A. Hendi, Influence of compliant

walls on peristaltic motion with heat mass transfer and chemical reaction,
International Journal of Heat and Mass Transfer, 55 (2012), 33863394.

[18] Nabil.T.Eldabe, kawther.A.Kamel, Galila.M.Abd-Allah and
Shaimaa.F.Ramadan Peristaltic motion with heat and mass transfer
of a dusty fluid through a horizontal porous channel under the effect of
wall properties, IJRRAS, 15 (2013), 300-311.

[19] P. Srinivasa Rao, G. Bhanodaya Reddy, Effect of reflux on peristaltic mo
tion in an asymmetric channel with partial slip and different wave forms,
International journal of mechanical engineering (IJME), 3 (2014), 35-48.

[20] Dheia G. Salih, Al-Khafajy and Ahmed M. Abdulhadi, Effects of wall

properties and heat transfer on the peristaltic transport of a Jeffrey fluid
through porous medium channel, Mathematical Theory and Modeling, 4
(2014), pp. 88-99.

[21] S. Nadeem, Arshad Riaz, R. Ellahi, N. S. Akbar, Mathematical model for

the peristaltic flow of Jeffrey fluid with nanoparticles phenomenon through
a rectangular duct, Appl Nanosci, 4 (2014), 613-624.

ANALYSIS ON THE EFFECTS OF WALL PROPERTIES ON... 263

[22] S. Nadeem, Arshad Riaz and R. Ellahi, Peristaltic flow of viscous fluid in

a rectangular duct with compliant walls, Computational Mathematics and
Modeling, 25 (2014), 404- 415.

[23] G.C.Sankad, P.S.Nagathan, Asha Patil, M.Y.Dhange, Peristaltic Trans
port of a Herschel-Bulkley Fluid in a Non-Uniform Channel with Wall
Effects, International Journal of Engineering Science and Innovative Technology, 3 (2014), 669-678.

264

본문 출처 PDF: PDF 보기 · 다른 PDF는 위 [PDF 업로드]로 올릴 수 있습니다.

활용가능성

이력 0건

아직 편집 이력이 없습니다. 저장할 때마다 이전 내용이 여기에 백업됩니다.